Notes: `a_0 = 3, a_1 = 3, a_4 = 15` Find a quadratic model for the sequence with the indicated terms.

Thursday, March 13, 2014

You need to remember what a quadratic model is, such that:

$\displaystyle{a}_{{n}}={f{{\left({n}\right)}}}={a}\cdot{{n}}^{{2}}+{b}\cdot{n}+{c}$

The problem provides the following information, such that:

$\displaystyle{a}_{{0}}={3}\Rightarrow{f{{\left({0}\right)}}}={a}\cdot{{0}}^{{2}}+{b}\cdot{0}+{c}\Rightarrow{c}={3}$

$\displaystyle{a}_{{1}}={3}\Rightarrow{f{{\left({1}\right)}}}={a}\cdot{{1}}^{{2}}+{b}\cdot{1}+{c}\Rightarrow{a}+{b}+{c}={3}$

$\displaystyle{a}_{{4}}={15}\Rightarrow{f{{\left({4}\right)}}}={a}\cdot{{4}}^{{2}}+{b}\cdot{4}+{c}\Rightarrow{16}{a}+{4}{b}+{c}={15}$

You need to replace 3 for c in equation $\displaystyle{a}+{b}+{c}={3}$ :

$\displaystyle{a}+{b}+{3}={3}\Rightarrow{a}+{b}={0}$

You need to replace 3 for c in equation $\displaystyle{16}{a}+{4}{b}+{c}={15}$ :

$\displaystyle{16}{a}+{4}{b}+{3}={15}\Rightarrow{16}{a}+{4}{b}={12}\Rightarrow{4}{a}+{b}={3}$

Subtract $\displaystyle{a}+{b}={0}$ from $\displaystyle{4}{a}+{b}={3}$ , such that:

$\displaystyle{4}{a}+{b}-{a}-{b}={3}$

$\displaystyle{3}{a}={3}\Rightarrow{a}={1}$

Replace 1 for a in equation $\displaystyle{a}+{b}={0}$ such that:

$\displaystyle{1}+{b}={0}\Rightarrow{b}=-{1}$

Hence, the quadratic model for the given sequence is $\displaystyle{a}_{{n}}={{n}}^{{2}}-{n}+{3}.$

Notes